Doping dependence of the nonlinear Hall resistivity in electron-doped Pr$_{2-x}$Ce$_{x}$CuO$_{4 \pm \delta}$

We report on a systematic study of the field dependence of the Hall resistivity $\rho _{yx} (B)$ as a function of doping in thin films of electron-doped superconducting cuprate Pr$_{2-x}$Ce$_{x}$CuO$_{4 \pm \delta}$ (PCCO). Across the studied doping range from $x = 0.125$ to $x = 0.20$, we observe a nonlinear dependence of $\rho_{yx}$ with $B$. The leading $B^{3}$ nonlinear term is negative, increases with decreasing temperature and peaks around optimal doping ($x = 0.15$). The observed nonlinear contribution is consistent with the presence of two different types of free carriers (electron-like and hole-like) even for doping with only an apparent hole Fermi surface as observed by angle-resolved photoemission spectroscopy. Based on an analysis using the two-carrier model, this negative nonlinear contribution to $\rho _{yx} (B)$ implies that the density of the charge carriers behaving like electrons is larger than the density of those behaving like holes for all the doping values explored. Combined with the Hall coefficient reaching $R_{H} = 0$ at specific temperatures for selected doping levels, we also conclude that the mobility of hole-like carriers is larger than that of electron-like carriers for doping around $x^{*} \approx 0.165$.